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Stochastic gradient descent algorithm for stochastic optimization in solving analytic continuation problems

  • * Corresponding author: Feng Bao

    * Corresponding author: Feng Bao 
The first author is supported by NSF grant DMS-1720222
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  • We propose a stochastic gradient descent based optimization algorithm to solve the analytic continuation problem in which we extract real frequency spectra from imaginary time Quantum Monte Carlo data. The procedure of analytic continuation is an ill-posed inverse problem which is usually solved by regularized optimization methods, such like the Maximum Entropy method, or stochastic optimization methods. The main contribution of this work is to improve the performance of stochastic optimization approaches by introducing a supervised stochastic gradient descent algorithm to solve a flipped inverse system which processes the random solutions obtained by a type of Fast and Efficient Stochastic Optimization Method.

    Mathematics Subject Classification: Primary: 49N45; Secondary: 49M37.


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  • Figure 1.  Example 1. True spectrum

    Figure 2.  Example 1. (a) FESOM samples; (b) FESOM estimation

    Figure 3.  Example 1. Estimated spectrum learned from FESOM samples

    Figure 4.  Example 2. True spectrum

    Figure 5.  Example 2. (a) FESOM estimation; (b) Estimated spectrum learned from FESOM samples

    Figure 6.  Example 2. Comparison between SGD and MaxEnt

    Figure 7.  True spectrum

    Figure 8.  Example 3. Estimations for the spectrum

    Figure 9.  Example 3. Spectrum with fine feature in positive frequency region

    Figure 10.  Example 3. (a) MaxEnt estimation for $ A_2 $; (b) Comparison of MaxEnt in estimating $ A_1 $ (red) and $ A_2 $ (blue)

    Figure 11.  Example 3. (a) SGD estimation for $ A_1 $; (b) SGD estimation for $ A_2 $

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